Approximate decoherence of histories and ’t Hooft’s deterministic quantum theory

In the decoherent histories approach to quantum theory, sets of histories are said to be decoherent when the decoherence functional, measuring interference between pairs of histories, is exactly diagonal. In realistic situations, however, only approximate diagonality is ever achieved, raising the question of what approximate decoherence actually means and how it is related to exact decoherence. This paper explores the possibility that an exactly decoherent set of histories may be constructed from an approximate set by small distortions of the operators characterizing the histories. In particular, for the case of histories of positions and momenta, this is achieved by doubling the set of operators and then finding, among this enlarged set, new position and momentum operators that commute, and so decohere exactly, and which are “close” to the original operators. Two derivations are given: one in terms of the decoherence functional, the second in terms of Wigner functions. The enlarged, exactly decoherent theory has the same classical dynamics as the original one, and coincides with the so-called deterministic quantum theories of the type recently studied by ’t Hooft. These results suggest that the comparison of standard and deterministic quantum theories may provide an alternative method of characterizing emergent classicality. A side product is the surprising result that histories of momenta in the quantum Brownian motion model (for the free particle in the high-temperature limit) are exactly decoherent.